It is in general very challenging to explicitly compute the Manolescu correction terms of a given \((Y,\mathfrak {s})\) , or even to understand its behavior under connected sums [1]}, [2]}, [3]}. Fortunately, in order to prove our main result we will need a much less refined understanding of these invariants. We begin by showing how to relate the \(\mathrm {Pin}(2)\) -width to the \(U\) -torsion. In particular, we show the following.
